Human languages share an optimal information transmission rate
Full statistical analysis and plots
Christophe Coupé, Dan Dediu and François Pellegrino
2019-03-18 16:57:12
Introduction
This Rmarkdown script (and corresponding TAB-separated CSV input data file InfoRateData.csv and the resulting HTML document) contain the full analysis and plotting code accompanying the paper Human languages share an optimal information transmission rate.
The data
For more information on the data, please see Oh (2015). There are in total 17 languages (see the Table below).
The Oral Corpus
The oral corpus is based on a subset of the Multext (Multilingual Text Tools and Corpora) parallel corpus (Campione & Véronis, 1998) in British English, German, and Italian. The material consists of 15 short texts of 3-5 semantically connected sentences carefully translated by a native speaker in each language.
For the other 14 languages, two of the authors supervised the translation and recording of new datasets. All participants were native speakers of the target language, with a focus on a specific variety of the language when possible – e.g. Mandarin spoken in Beijing, Serbian in Belgrade and Korean in Seoul. No strict control on age or on the speakers’ social diversity was performed, but speakers were mainly students or members of academic institutions. Speakers were asked to read three times (first silently and then loudly twice) each text. The texts were presented one by one on the screen in random order, in a self-paced reading paradigm. This way, speakers familiarized themselves with the text and reduce their reading errors. The second loud recording was analyzed in this study.
Text Corpus
Text datasets were acquired from various sources as illustrated in the Table below. After an initial data curation, each dataset was phonetically transcribed and automatically syllabified by a rule-based program written by one of the authors, except in the following cases:
- when syllabification was already provided with the dataset (English, French, German, and Vietnamese for the multisyllabic words);
- when the corpus was syllabified by an automatic grapheme-to-phoneme converter (Catalan, Spanish, and Thai).
Additionally, no syllabification was required for Sino-Tibetan languages (Cantonese and Mandarin Chinese) since one ideogram corresponds to one syllable.
| Language | Family | ISO 639-3 | Corpus |
|---|---|---|---|
| Basque | Basque | EUS | E-Hitz (Perea et al., 2006) |
| British English | Indo-European | ENG | WebCelex (MPI for Psycholinguistics) |
| Cantonese | Sino-Tibetan | YUE | A linguistic corpus of mid-20th century Hong Kong Cantonese |
| Catalan | Indo-European | CAT | Frequency dictionary (Zséder et al., 2012) |
| Finnish | Uralic | FIN | Finnish Parole Corpus |
| French | Indo-European | FRA | Lexique 3.80 (New et al., 2001) |
| German | Indo-European | DEU | WebCelex (MPI for Psycholinguistics) |
| Hungarian | Uralic | HUN | Hungarian National Corpus (Váradi, 2002) |
| Italian | Indo-European | ITA | The Corpus PAISÀ (Lyding et al., 2014) |
| Japanese | Japanese | JPN | Japanese Internet Corpus (Sharoff, 2006) |
| Korean | Korean | KOR | Leipzig Corpora Collection (LCC) |
| Mandarin Chinese | Sino-Tibetan | CMN | Chinese Internet Corpus (Sharoff, 2006) |
| Serbian | Indo-European | SRP | Frequency dictionary (Zséder et al., 2012) |
| Spanish | Indo-European | SPA | Frequency dictionary (Zséder et al., 2012) |
| Thai | Tai-Kadai | THA | Thai National Corpus (TNC) |
| Turkish | Turkic | TUR | Leipzig Corpora Collection (LCC) |
| Vietnamese | Austroasiatic | VIE | VNSpeechCorpus (Le et al., 2004) |
Dataset structure
The data is structured as follows:
- Language is the unique language ISO 639-3 ID with 17 possible values “CAT”, “CMN”, “DEU”, “ENG”, “EUS”, “FIN”, “FRA”, “HUN”, “ITA”, “JPN”, “KOR”, “SPA”, “SRP”, “THA”, “TUR”, “VIE”, “YUE”;
- Language family is the language family with 9 possible values “Austroasiatic”, “Basque”, “Indo-European”, “Japanese”, “Korean”, “Sino-Tibetan”, “Tai-Kadai”, “Turkic”, “Uralic”;
- Text is the text identifier (similar across languages and speakers) with 15 possible values “O1”, “O2”, “O3”, “O4”, “O6”, “O8”, “O9”, “P0”, “P1”, “P2”, “P3”, “P8”, “P9”, “Q0”, “Q1”;
- Speaker is the unique speaker’s ID (there are 170 in the dataset);
- Sex is the speaker’s sex;
- Age is the speaker’s age (available only for a subset of 132 speakers);
- Duration is the texts’ duration in seconds as spoken by the given speaker (pauses longer than 150ms being excluded);
- NS is the texts’ number of of syllables (the same for a given text in a given language);
- SR is the speech rate (syllables/second) for a given text spoken by a given speaker;
- ShE (resp. ID) is the first-order (resp. second-order) language-level entropy estimate;
- ShIR and IR are text * speaker properties obtained by multiplying SR by ShE and ID respectively.
Technical notes on regressions
We use throughout sum contrasts for the factor IVs, which are orthogonal contrasts which compare every level of the IV to the overall mean (for example, for a two-levels factor such as Sex we do not compare Males with Females but each with their overall mean, which is included in the intercept). However, in R the contr.sum() function used to define this contrasts produces level names that are very uninformative, so we explicit these below (please note that in the model outputs the last level is usually not shown):
- Sex:
Sex1= F,Sex2= M (the last level,Sex2is usually not displayed); - Text:
Text1= O1,Text2= O2,Text3= O3,Text4= O4,Text5= O6,Text6= O8,Text7= O9,Text8= P0,Text9= P1,Text10= P2,Text11= P3,Text12= P8,Text13= P9,Text14= Q0,Text15= Q1 (the last level,Text15is usually not displayed); - Language:
Language1= CAT,Language2= CMN,Language3= DEU,Language4= ENG,Language5= EUS,Language6= FIN,Language7= FRA,Language8= HUN,Language9= ITA,Language10= JPN,Language11= KOR,Language12= SPA,Language13= SRP,Language14= THA,Language15= TUR,Language16= VIE,Language17= YUE (the last level,Language17is usually not displayed); - Family:
Family1= Austroasiatic,Family2= Basque,Family3= Indo-European,Family4= Japanese,Family5= Korean,Family6= Sino-Tibetan,Family7= Tai-Kadai,Family8= Turkic,Family9= Uralic (the last level,Family9is usually not displayed);
Exploratory plots and summaries
Speaker characteristics
| Lng | # spkrs | % fem | # age | mean(age) | sd(age) | actual ages |
|---|---|---|---|---|---|---|
| CAT | 10 | 50 | 10 | 35.4 | 9.2 | (21, 28, 28, 29, 31, 39, 42, 42, 44, 50) |
| CMN | 10 | 50 | 9 | 23.1 | 4.5 | (19, 19, 19, 19, 24, 24, 25, 28, 31) |
| DEU | 10 | 50 | 0 | NaN | NaN | () |
| ENG | 10 | 50 | 0 | NaN | NaN | () |
| EUS | 10 | 50 | 10 | 28.0 | 4.9 | (19, 22, 26, 27, 28, 29, 30, 31, 32, 36) |
| FIN | 10 | 50 | 10 | 33.2 | 11.0 | (16, 22, 26, 28, 30, 35, 37, 41, 45, 52) |
| FRA | 10 | 50 | 10 | 32.5 | 7.7 | (24, 25, 25, 27, 28, 36, 36, 37, 41, 46) |
| HUN | 10 | 50 | 10 | 39.3 | 15.8 | (17, 27, 27, 31, 33, 39, 42, 51, 57, 69) |
| ITA | 10 | 50 | 0 | NaN | NaN | () |
| JPN | 10 | 50 | 10 | 30.6 | 12.8 | (20, 20, 21, 22, 22, 28, 29, 40, 51, 53) |
| KOR | 10 | 50 | 10 | 28.6 | 10.6 | (16, 19, 19, 19, 28, 31, 33, 35, 36, 50) |
| SPA | 10 | 50 | 10 | 33.7 | 10.1 | (21, 22, 26, 28, 30, 32, 42, 42, 44, 50) |
| SRP | 10 | 50 | 10 | 30.6 | 7.8 | (19, 21, 23, 30, 31, 32, 34, 34, 38, 44) |
| THA | 10 | 50 | 10 | 30.1 | 5.7 | (23, 23, 27, 28, 30, 31, 31, 32, 33, 43) |
| TUR | 10 | 50 | 7 | 32.6 | 7.2 | (24, 25, 30, 31, 37, 37, 44) |
| VIE | 10 | 50 | 6 | 27.2 | 4.1 | (21, 25, 26, 28, 31, 32) |
| YUE | 10 | 50 | 10 | 22.0 | 1.5 | (20, 20, 21, 21, 22, 22, 23, 23, 24, 24) |
NS
NS: exploratory plots.
mean=101.196, median=100, sd=24.703, CV=0.244, min=49, max=162, kurtosis=2.484, skewness=0.177.SR
SR: exploratory plots.
SR per speaker.
SR by Sex and Age across Languages.
SR by Sex, Age and Language.
SR by language.
mean=6.631, median=6.777, sd=1.148, CV=0.173, min=3.589, max=9.492, kurtosis=2.408, skewness=-0.168.ShE and ID
ShE and ID: exploratory plots.
ShE vs ID.
Pearson's product-moment correlation
data: tmp1$ShE and tmp1$ID
t = 2.0326, df = 15, p-value = 0.06019
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
-0.02052914 0.77274779
sample estimates:
cor
0.4647009
Spearman's rank correlation rho
data: tmp1$ShE and tmp1$ID
S = 451.88, p-value = 0.07259
alternative hypothesis: true rho is not equal to 0
sample estimates:
rho
0.4462208
Paired t-test
data: tmp1$ShE and tmp1$ID
t = 11.635, df = 16, p-value = 3.213e-09
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
2.158040 3.119607
sample estimates:
mean of the differences
2.638824 ShE:
mean=8.621, median=8.69, sd=0.904, CV=0.105, min=6.07, max=9.83, kurtosis=4.665, skewness=-1.122.ID:
mean=6.009, median=5.56, sd=0.883, CV=0.147, min=4.83, max=8.02, kurtosis=2.53, skewness=0.747.ShIR and IR
ShIR: exploratory plots.
ShIR per speaker.
ShIR by Sex and Age across Languages.
ShIR by Sex, Age and Language.
ShIR by language.
IR: exploratory plots.
IR per speaker.
IR by Sex and Age across Languages.
IR by Sex, Age and Language.
IR by language.
ShIR:
mean=56.709, median=57.207, sd=9.35, CV=0.165, min=32.772, max=89.235, kurtosis=2.444, skewness=0.079.IR:
mean=39.153, median=39.13, sd=5.097, CV=0.13, min=25.631, max=60.692, kurtosis=3.622, skewness=0.325.SR and ID
SR vs ID
Pearson's product-moment correlation
data: info.rate.data$SR and info.rate.data$ID
t = -45.329, df = 2286, p-value < 2.2e-16
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
-0.7090031 -0.6658066
sample estimates:
cor
-0.6880138
Spearman's rank correlation rho
data: info.rate.data$SR and info.rate.data$ID
S = 3393600000, p-value < 2.2e-16
alternative hypothesis: true rho is not equal to 0
sample estimates:
rho
-0.6999614 Linear Mixed Modells (LMMs)
Linear ICCs
| Level-1 factor (f) | ICC |
|---|---|
| Text | 0.26 |
| Family | 0.42 |
| Language | 0.19 |
| Speaker | 0.00 |
| Level-1 factor (f) | ICC |
|---|---|
| Text | 0.01 |
| Family | 0.50 |
| Language | 0.19 |
| Speaker | 0.24 |
| Level-1 factor (f) | ICC |
|---|---|
| Text | 0.01 |
| Family | 0.47 |
| Language | 0.13 |
| Speaker | 0.30 |
| Level-1 factor (f) | ICC |
|---|---|
| Text | 0.02 |
| Family | 0.03 |
| Language | 0.32 |
| Speaker | 0.49 |
SR
Comparing models
| model | AIC | BIC |
|---|---|---|
| 1 + (1 | Text) + (1 | Family/Language) + (1 | Speaker) | 2127.51 | 2161.93 |
| 1 + (1 | Family/Language) + (1 | Speaker) | 2397.97 | 2426.65 |
| 1 + (1 | Text) + (1 | Speaker) | 2257.73 | 2280.68 |
| 1 + (1 | Text) + (1 | Family/Language) | 4717.06 | 4745.73 |
| 1 + Sex + (1 | Text) + (1 | Family/Language) + (1 | Speaker) | 2121.04 | 2161.19 |
| 1 + Sex + (1 | Family/Language) + (1 | Speaker) | 2391.29 | 2425.7 |
| 1 + Sex + (1 | Text) + (1 | Speaker) | 2258.63 | 2287.31 |
| 1 + Sex + (1 | Text) + (1 | Family/Language) | 4584.11 | 4618.52 |
Residuals and random effects
We consider here the full model SR ~ 1 + Sex + (1|Text) + (1|Family/Language) + (1|Speaker).
IR
Comparing models
| model | AIC | BIC |
|---|---|---|
| 1 + (1 | Text) + (1 | Family/Language) + (1 | Speaker) | 10257.84 | 10292.25 |
| 1 + (1 | Family/Language) + (1 | Speaker) | 10524.5 | 10553.18 |
| 1 + (1 | Text) + (1 | Speaker) | 10305.76 | 10328.71 |
| 1 + (1 | Text) + (1 | Family/Language) | 12888.49 | 12917.17 |
| 1 + Sex + (1 | Text) + (1 | Family/Language) + (1 | Speaker) | 10247.34 | 10287.49 |
| 1 + Sex + (1 | Family/Language) + (1 | Speaker) | 10513.79 | 10548.21 |
| 1 + Sex + (1 | Text) + (1 | Speaker) | 10300.05 | 10328.72 |
| 1 + Sex + (1 | Text) + (1 | Family/Language) | 12747.04 | 12781.45 |
Residuals and random effects
We consider here the full model IR ~ 1 + Sex + (1|Text) + (1|Family/Language) + (1|Speaker).
Generalized Additive Models for Location, Scale and Shape (GAMLSS)
We will use a Gaussian distribution (with fixed or modelled variance).
SR
Fixed or modelled σ (variance)
Fixed σ
******************************************************************
Summary of the Quantile Residuals
mean = -7.131581e-05
variance = 1.000437
coef. of skewness = 0.04303609
coef. of kurtosis = 3.710291
Filliben correlation coefficient = 0.9979187
******************************************************************
Deviance= 1177.818
AIC= 1576.98 Modelled σ
******************************************************************
Summary of the Quantile Residuals
mean = 0.002001851
variance = 1.000433
coef. of skewness = 0.03014483
coef. of kurtosis = 2.864355
Filliben correlation coefficient = 0.9994151
******************************************************************
Deviance= 815.9172
AIC= 1405.705 The distribution of the residuals is less heteroscedastic than before and the fit to the data better. The full summary of the model is:
******************************************************************
Family: c("NO", "Normal")
Call: gamlss(formula = SR ~ 1 + Sex + random(Text) + random(Language) + random(Family) + random(Speaker), sigma.formula = ~1 + Sex + random(Text) + random(Language) + random(Family) +
random(Speaker), family = NO(mu.link = "identity"), data = d, control = gamlss.control(n.cyc = 800, trace = FALSE), i.control = glim.control(bf.cyc = 800))
Fitting method: RS()
------------------------------------------------------------------
Mu link function: identity
Mu Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 6.629552 0.005816 1139.83 <2e-16 ***
Sex1 -0.168157 0.005816 -28.91 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------------
Sigma link function: log
Sigma Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.23693 0.01478 -83.673 < 2e-16 ***
Sex1 -0.05788 0.01478 -3.915 9.34e-05 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------------
NOTE: Additive smoothing terms exist in the formulas:
i) Std. Error for smoothers are for the linear effect only.
ii) Std. Error for the linear terms maybe are not accurate.
------------------------------------------------------------------
No. of observations in the fit: 2288
Degrees of Freedom for the fit: 294.8937
Residual Deg. of Freedom: 1993.106
at cycle: 53
Global Deviance: 815.9172
AIC: 1405.705
SBC: 3097.048
******************************************************************Random effects
μ
Text
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 14.39185
Random effect parameter sigma_b: 0.109103
Smoothing parameter lambda : 84.5402 Language
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 16.98344
Random effect parameter sigma_b: 0.870621
Smoothing parameter lambda : 1.32916 Family
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 0.01237646
Random effect parameter sigma_b: 1.91981e-05
Smoothing parameter lambda : 2713240000 Speaker
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 165.4211
Random effect parameter sigma_b: 0.569042
Smoothing parameter lambda : 3.32891 σ
Text
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 0.00152528
Random effect parameter sigma_b: 0.000518285
Smoothing parameter lambda : 3475370 Language
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 14.62296
Random effect parameter sigma_b: 0.153319
Smoothing parameter lambda : 39.9698 Family
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 0.001713103
Random effect parameter sigma_b: 0.000184844
Smoothing parameter lambda : 27323100 Speaker
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 79.45882
Random effect parameter sigma_b: 0.181484
Smoothing parameter lambda : 29.3639 IR
Fixed or modelled σ (variance)
Fixed σ
******************************************************************
Summary of the Quantile Residuals
mean = 8.557955e-05
variance = 1.000437
coef. of skewness = 0.09721211
coef. of kurtosis = 3.684191
Filliben correlation coefficient = 0.9978017
******************************************************************
Deviance= 9322.636
AIC= 9721.861 Modelled σ
******************************************************************
Summary of the Quantile Residuals
mean = 0.002104815
variance = 1.000442
coef. of skewness = 0.03139581
coef. of kurtosis = 2.864031
Filliben correlation coefficient = 0.9993924
******************************************************************
Deviance= 8961.553
AIC= 9554.321 Again, this is a better fit to the data. The full summary of the model is:
******************************************************************
Family: c("NO", "Normal")
Call: gamlss(formula = IR ~ 1 + Sex + random(Text) + random(Language) + random(Family) + random(Speaker), sigma.formula = ~1 + Sex + random(Text) + random(Language) + random(Family) +
random(Speaker), family = NO(mu.link = "identity"), data = d, control = gamlss.control(n.cyc = 800, trace = FALSE), i.control = glim.control(bf.cyc = 800))
Fitting method: RS()
------------------------------------------------------------------
Mu link function: identity
Mu Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 39.14451 0.03463 1130.31 <2e-16 ***
Sex1 -1.01064 0.03463 -29.18 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------------
Sigma link function: log
Sigma Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.54554 0.01478 36.903 < 2e-16 ***
Sex1 -0.05935 0.01478 -4.015 6.17e-05 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------------
NOTE: Additive smoothing terms exist in the formulas:
i) Std. Error for smoothers are for the linear effect only.
ii) Std. Error for the linear terms maybe are not accurate.
------------------------------------------------------------------
No. of observations in the fit: 2288
Degrees of Freedom for the fit: 296.3836
Residual Deg. of Freedom: 1991.616
at cycle: 6
Global Deviance: 8961.553
AIC: 9554.321
SBC: 11254.21
******************************************************************Random effects
μ
Text
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 14.41202
Random effect parameter sigma_b: 0.660353
Smoothing parameter lambda : 2.30777 Language
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 16.95235
Random effect parameter sigma_b: 3.10086
Smoothing parameter lambda : 0.104779 Family
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 0.01908253
Random effect parameter sigma_b: 0.000233421
Smoothing parameter lambda : 18354000 Speaker
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 165.2917
Random effect parameter sigma_b: 3.39374
Smoothing parameter lambda : 0.0935855 σ
Text
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 0.0004688834
Random effect parameter sigma_b: 0.000282878
Smoothing parameter lambda : 11664800 Language
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 14.73159
Random effect parameter sigma_b: 0.157607
Smoothing parameter lambda : 37.8208 Family
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 0.001453389
Random effect parameter sigma_b: 0.00011236
Smoothing parameter lambda : 73935500 Speaker
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 80.97495
Random effect parameter sigma_b: 0.184921
Smoothing parameter lambda : 28.2979 Modelling the relationship between SR and ID
Let’s model SR with ID as an additional predictor (fixed effect) interacting with Sex. N.B. In this case, we must drop Language as a random effect, since each language has, by definition, only one value of ID.
******************************************************************
Family: c("NO", "Normal")
Call: gamlss(formula = SR ~ 1 + ID * Sex + random(Text) + random(Speaker) + random(Family), sigma.formula = ~1 + ID + Sex + random(Text) + random(Speaker) + random(Family),
family = NO(mu.link = "identity"), data = d, control = gamlss.control(n.cyc = 800, trace = FALSE), i.control = glim.control(bf.cyc = 800))
Fitting method: RS()
------------------------------------------------------------------
Mu link function: identity
Mu Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 11.970094 0.037525 318.987 < 2e-16 ***
ID -0.888703 0.005992 -148.324 < 2e-16 ***
Sex1 -0.062504 0.037524 -1.666 0.09593 .
ID:Sex1 -0.017079 0.005991 -2.851 0.00441 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------------
Sigma link function: log
Sigma Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.73603 0.10289 -7.153 1.18e-12 ***
ID -0.08460 0.01694 -4.993 6.45e-07 ***
Sex1 -0.05723 0.01478 -3.871 0.000112 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------------
NOTE: Additive smoothing terms exist in the formulas:
i) Std. Error for smoothers are for the linear effect only.
ii) Std. Error for the linear terms maybe are not accurate.
------------------------------------------------------------------
No. of observations in the fit: 2288
Degrees of Freedom for the fit: 293.3745
Residual Deg. of Freedom: 1994.626
at cycle: 8
Global Deviance: 790.2025
AIC: 1376.951
SBC: 3059.581
******************************************************************
******************************************************************
Summary of the Quantile Residuals
mean = 0.001905342
variance = 1.000433
coef. of skewness = 0.01947475
coef. of kurtosis = 2.77394
Filliben correlation coefficient = 0.9993355
******************************************************************
Deviance= 790.2025
AIC= 1376.951 Adding ID as a predictor improves the fits (as judged by AIC). There is a negative estimate for ID, but significance is difficult to assess with GAMLSS model involving smoothing functions. However, also using a simple lmer model we have a significant effect of ID:
Linear mixed model fit by REML. t-tests use Satterthwaite's method ['lmerModLmerTest']
Formula: SR ~ 1 + ID * Sex + (1 | Text) + (1 | Speaker) + (1 | Family)
Data: info.rate.data
REML criterion at convergence: 2121.2
Scaled residuals:
Min 1Q Median 3Q Max
-3.7806 -0.6226 0.0176 0.5808 5.1707
Random effects:
Groups Name Variance Std.Dev.
Speaker (Intercept) 0.4618 0.6795
Text (Intercept) 0.0172 0.1311
Family (Intercept) 0.2443 0.4942
Residual 0.1063 0.3260
Number of obs: 2288, groups: Speaker, 170; Text, 15; Family, 9
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 10.521343 0.637420 42.244565 16.506 < 2e-16 ***
ID -0.658948 0.101261 54.414768 -6.507 2.52e-08 ***
Sex1 -0.128232 0.367323 155.446204 -0.349 0.727
ID:Sex1 -0.006959 0.060377 155.565771 -0.115 0.908
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) ID Sex1
ID -0.959
Sex1 0.002 -0.002
ID:Sex1 -0.002 0.002 -0.990Type III Analysis of Variance Table with Satterthwaite's method
Sum Sq Mean Sq NumDF DenDF F value Pr(>F)
ID 4.4998 4.4998 1 54.415 42.3465 2.516e-08 ***
Sex 0.0130 0.0130 1 155.446 0.1219 0.7275
ID:Sex 0.0014 0.0014 1 155.566 0.0133 0.9084
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Random effects
μ
Text
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 14.41708
Random effect parameter sigma_b: 0.11065
Smoothing parameter lambda : 82.1944 Speaker
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 167.4872
Random effect parameter sigma_b: 0.7577
Smoothing parameter lambda : 1.8794 Family
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 0.01763365
Random effect parameter sigma_b: 0.000168856
Smoothing parameter lambda : 35072900 σ
Text
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 0.03217714
Random effect parameter sigma_b: 0.00241144
Smoothing parameter lambda : 152754 Speaker
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 104.4197
Random effect parameter sigma_b: 0.242522
Smoothing parameter lambda : 15.8243 Family
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 0.000711841
Random effect parameter sigma_b: 0.000279392
Smoothing parameter lambda : 11379200 Do Age and Sex matter?
Linear mixed model fit by REML. t-tests use Satterthwaite's method ['lmerModLmerTest']
Formula: SR ~ Age * Sex + (1 | Text) + (1 | Language)
Data: info.rate.data
REML criterion at convergence: 3743.7
Scaled residuals:
Min 1Q Median 3Q Max
-3.4947 -0.6305 0.0105 0.5978 3.6970
Random effects:
Groups Name Variance Std.Dev.
Text (Intercept) 0.01299 0.1140
Language (Intercept) 1.01328 1.0066
Residual 0.36283 0.6024
Number of obs: 1979, groups: Text, 15; Language, 14
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 6.809e+00 2.753e-01 1.417e+01 24.734 4.61e-13 ***
Age -5.972e-03 1.592e-03 1.951e+03 -3.751 0.000181 ***
Sex1 -1.084e-01 4.608e-02 1.948e+03 -2.351 0.018804 *
Age:Sex1 -1.223e-03 1.440e-03 1.949e+03 -0.849 0.395965
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) Age Sex1
Age -0.176
Sex1 0.018 -0.107
Age:Sex1 -0.019 0.115 -0.956Linear mixed model fit by REML. t-tests use Satterthwaite's method ['lmerModLmerTest']
Formula: IR ~ Age * Sex + (1 | Text) + (1 | Family/Language)
Data: info.rate.data
REML criterion at convergence: 10792.3
Scaled residuals:
Min 1Q Median 3Q Max
-3.2256 -0.6274 0.0068 0.5901 4.6601
Random effects:
Groups Name Variance Std.Dev.
Text (Intercept) 0.4338 0.6586
Language:Family (Intercept) 8.4982 2.9152
Family (Intercept) 2.1402 1.4629
Residual 12.9843 3.6034
Number of obs: 1979, groups: Text, 15; Language:Family, 14; Family, 9
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 3.996e+01 1.009e+00 1.049e+01 39.596 9.13e-13 ***
Age -3.985e-02 9.517e-03 1.954e+03 -4.187 2.95e-05 ***
Sex1 -6.705e-01 2.756e-01 1.950e+03 -2.433 0.0151 *
Age:Sex1 -7.286e-03 8.615e-03 1.950e+03 -0.846 0.3978
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) Age Sex1
Age -0.285
Sex1 0.028 -0.106
Age:Sex1 -0.030 0.114 -0.956So, it seems Age and Sex are both worth including in our models (even if we have to discard quite a bit of data because of missing Age info). (In fact, the effect of Age seems more significant than that of Sex.)
In the following, we investigate if Age does matter when using GAMLSS modelling…
GAMLSS with Age
Bescause there is missing data fro Age, and because the GAMLSS models require no missing data, we will fit the models with Age (and its interaction with Sex) on the subset of the data that contains only those speakers with Age info. To make comparability possible, we also fit the same models but without Age on the exact same subset of the data.
SR
Fixed or modelled σ (variance)
Fixed σ
******************************************************************
Summary of the Quantile Residuals
mean = -8.252732e-05
variance = 1.000506
coef. of skewness = 0.04585
coef. of kurtosis = 3.821224
Filliben correlation coefficient = 0.9973638
******************************************************************
The model including Age * Sex is:
******************************************************************
Family: c("NO", "Normal")
Call: gamlss(formula = SR ~ 1 + Sex * Age + random(Text) + random(Language) + random(Family) + random(Speaker), family = NO(mu.link = "identity"), data = info.rate.data.for.age, control = gamlss.control(n.cyc = 800, trace = FALSE),
i.control = glim.control(bf.cyc = 800))
Fitting method: RS()
------------------------------------------------------------------
Mu link function: identity
Mu Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 6.7419779 0.0234704 287.254 < 2e-16 ***
Sex1 -0.0916878 0.0234704 -3.907 9.71e-05 ***
Age -0.0024831 0.0007319 -3.393 0.000707 ***
Sex1:Age -0.0017244 0.0007319 -2.356 0.018566 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------------
Sigma link function: log
Sigma Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.1605 0.0159 -73.01 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------------
NOTE: Additive smoothing terms exist in the formulas:
i) Std. Error for smoothers are for the linear effect only.
ii) Std. Error for the linear terms maybe are not accurate.
------------------------------------------------------------------
No. of observations in the fit: 1979
Degrees of Freedom for the fit: 161.6924
Residual Deg. of Freedom: 1817.308
at cycle: 35
Global Deviance: 1022.798
AIC: 1346.183
SBC: 2250.099
******************************************************************The compared models are:
| Model | Deviance | AIC |
|---|---|---|
| Age * Sex | 1022.8 | 1346.2 |
| Age + Sex | 1022.8 | 1344.2 |
| Sex | 1022.8 | 1342.2 |
So, even if Age has a significant (negative) effect and interaction with Sex (positive for males), adding it does not seem to be warranted here…
Modelled σ
******************************************************************
Summary of the Quantile Residuals
mean = 0.002353284
variance = 1.000499
coef. of skewness = 0.02299299
coef. of kurtosis = 2.90592
Filliben correlation coefficient = 0.9993823
******************************************************************
The model including Age * Sex is:
******************************************************************
Family: c("NO", "Normal")
Call: gamlss(formula = SR ~ 1 + Sex * Age + random(Text) + random(Language) + random(Family) + random(Speaker), sigma.formula = ~1 + Sex * Age + random(Text) + random(Language) + random(Family) + random(Speaker),
family = NO(mu.link = "identity"), data = info.rate.data.for.age, control = gamlss.control(n.cyc = 800, trace = FALSE), i.control = glim.control(bf.cyc = 800))
Fitting method: RS()
------------------------------------------------------------------
Mu link function: identity
Mu Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 6.5465239 0.0203601 321.536 < 2e-16 ***
Sex1 -0.0442321 0.0203601 -2.172 0.03 *
Age 0.0038165 0.0006483 5.887 4.71e-09 ***
Sex1:Age -0.0030913 0.0006483 -4.768 2.01e-06 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------------
Sigma link function: log
Sigma Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.330351 0.053970 -24.650 <2e-16 ***
Sex1 0.019846 0.053970 0.368 0.7131
Age 0.003029 0.001685 1.798 0.0724 .
Sex1:Age -0.002561 0.001685 -1.520 0.1286
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------------
NOTE: Additive smoothing terms exist in the formulas:
i) Std. Error for smoothers are for the linear effect only.
ii) Std. Error for the linear terms maybe are not accurate.
------------------------------------------------------------------
No. of observations in the fit: 1979
Degrees of Freedom for the fit: 243.8639
Residual Deg. of Freedom: 1735.136
at cycle: 10
Global Deviance: 721.7861
AIC: 1209.514
SBC: 2572.798
******************************************************************The compared models are:
| Model | Deviance | AIC |
|---|---|---|
| Age * Sex | 721.8 | 1209.5 |
| Age + Sex | 721.5 | 1206.1 |
| Sex | 721.3 | 1203 |
So, even if Age has a significant (negative) effect (but no interaction with Sex), adding it does not seem to be warranted here either…
The distribution of the residuals is less heteroscedastic than before and the fit to the data better.
Summary
Thus, for SR, even if there is a hint that Age might affect it negatively (and there might also be an interaction with Sex with a positive effect for males), overall, the various fit indices do not warrant its inclusion in the GAMLSS models.
IR
Fixed or modelled σ (variance)
Fixed σ
******************************************************************
Summary of the Quantile Residuals
mean = 1.384527e-05
variance = 1.000506
coef. of skewness = 0.09166289
coef. of kurtosis = 3.764844
Filliben correlation coefficient = 0.9974381
******************************************************************
The model including Age * Sex is:
******************************************************************
Family: c("NO", "Normal")
Call: gamlss(formula = IR ~ 1 + Sex * Age + random(Text) + random(Language) + random(Family) + random(Speaker), family = NO(mu.link = "identity"), data = info.rate.data.for.age, control = gamlss.control(n.cyc = 800, trace = FALSE),
i.control = glim.control(bf.cyc = 800))
Fitting method: RS()
------------------------------------------------------------------
Mu link function: identity
Mu Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 40.015135 0.137507 291.003 < 2e-16 ***
Sex1 -0.175295 0.137508 -1.275 0.203
Age -0.035742 0.004288 -8.336 < 2e-16 ***
Sex1:Age -0.023187 0.004288 -5.408 7.22e-08 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------------
Sigma link function: log
Sigma Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.6074 0.0159 38.21 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------------
NOTE: Additive smoothing terms exist in the formulas:
i) Std. Error for smoothers are for the linear effect only.
ii) Std. Error for the linear terms maybe are not accurate.
------------------------------------------------------------------
No. of observations in the fit: 1979
Degrees of Freedom for the fit: 161.827
Residual Deg. of Freedom: 1817.173
at cycle: 2
Global Deviance: 8020.303
AIC: 8343.957
SBC: 9248.626
******************************************************************The compared models are:
| Model | Deviance | AIC |
|---|---|---|
| Age * Sex | 8020.3 | 8344 |
| Age + Sex | 8020.3 | 8342 |
| Sex | 8020.3 | 8340 |
So, even if Age has a significant (negative) effect and interaction with Sex (positive for males) – interestingly, in this case the main effect of Sex disappears –, adding it does not seem to be warranted…
Modelled σ
******************************************************************
Summary of the Quantile Residuals
mean = 0.002387788
variance = 1.00051
coef. of skewness = 0.02686615
coef. of kurtosis = 2.9163
Filliben correlation coefficient = 0.9993496
******************************************************************
The model including Age * Sex is:
******************************************************************
Family: c("NO", "Normal")
Call: gamlss(formula = IR ~ 1 + Sex * Age + random(Text) + random(Language) + random(Family) + random(Speaker), sigma.formula = ~1 + Sex * Age + random(Text) + random(Language) + random(Family) + random(Speaker),
family = NO(mu.link = "identity"), data = info.rate.data.for.age, control = gamlss.control(n.cyc = 800, trace = FALSE), i.control = glim.control(bf.cyc = 800))
Fitting method: RS()
------------------------------------------------------------------
Mu link function: identity
Mu Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 40.041563 0.118455 338.031 < 2e-16 ***
Sex1 -0.299328 0.118455 -2.527 0.0116 *
Age -0.036894 0.003708 -9.950 < 2e-16 ***
Sex1:Age -0.019096 0.003708 -5.150 2.9e-07 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------------
Sigma link function: log
Sigma Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.479445 0.053990 8.880 <2e-16 ***
Sex1 0.025025 0.053990 0.464 0.643
Age 0.001854 0.001685 1.100 0.271
Sex1:Age -0.002747 0.001685 -1.630 0.103
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------------
NOTE: Additive smoothing terms exist in the formulas:
i) Std. Error for smoothers are for the linear effect only.
ii) Std. Error for the linear terms maybe are not accurate.
------------------------------------------------------------------
No. of observations in the fit: 1979
Degrees of Freedom for the fit: 244.3689
Residual Deg. of Freedom: 1734.631
at cycle: 10
Global Deviance: 7726.653
AIC: 8215.391
SBC: 9581.498
******************************************************************The compared models are:
| Model | Deviance | AIC |
|---|---|---|
| Age * Sex | 7726.7 | 8215.4 |
| Age + Sex | 7726.4 | 8212.2 |
| Sex | 7725.8 | 8208.8 |
So, even if Age has a significant (negative) effect and interaction with Sex (positive for males) – interestingly, in this case the main effect of Sex disappears –, adding it does not seem to be warranted…
The distribution of the residuals is less heteroscedastic than before and the fit to the data better.
Summary
Thus, while for IR the hint that Age has a negative main effect and interacts with Sex (with a positive effect for males, containing the whole effect of Sex) is much stronger, the various fit indices do not warrant its inclusion in the GAMLSS models.
Modelling the relationship between SR and ID
******************************************************************
Summary of the Quantile Residuals
mean = 0.001744301
variance = 1.000502
coef. of skewness = 0.008063776
coef. of kurtosis = 2.834364
Filliben correlation coefficient = 0.9993747
******************************************************************
The model including Age * Sex is:
******************************************************************
Family: c("NO", "Normal")
Call: gamlss(formula = SR ~ 1 + ID + Sex * Age + random(Text) + random(Speaker) + random(Family), sigma.formula = ~1 + ID + Sex * Age + random(Text) + random(Speaker) + random(Family), family = NO(mu.link = "identity"),
data = info.rate.data.for.age, control = gamlss.control(n.cyc = 800, trace = FALSE), i.control = glim.control(bf.cyc = 800))
Fitting method: RS()
------------------------------------------------------------------
Mu link function: identity
Mu Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 12.6793628 0.0517065 245.218 < 2e-16 ***
ID -0.9813533 0.0071683 -136.903 < 2e-16 ***
Sex1 0.0102517 0.0204390 0.502 0.616
Age -0.0059906 0.0006663 -8.991 < 2e-16 ***
Sex1:Age -0.0050056 0.0006537 -7.658 3.12e-14 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------------
Sigma link function: log
Sigma Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.790248 0.133246 -5.931 3.63e-09 ***
ID -0.086078 0.019128 -4.500 7.24e-06 ***
Sex1 0.046494 0.054368 0.855 0.3926
Age 0.001961 0.001718 1.141 0.2540
Sex1:Age -0.003459 0.001698 -2.037 0.0418 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------------
NOTE: Additive smoothing terms exist in the formulas:
i) Std. Error for smoothers are for the linear effect only.
ii) Std. Error for the linear terms maybe are not accurate.
------------------------------------------------------------------
No. of observations in the fit: 1979
Degrees of Freedom for the fit: 236.269
Residual Deg. of Freedom: 1742.731
at cycle: 8
Global Deviance: 705.2126
AIC: 1177.751
SBC: 2498.577
******************************************************************The compared models are:
| Model | Deviance | AIC |
|---|---|---|
| ID * Sex * Age | 705.2 | 1188.4 |
| ID + Sex * Age | 705.2 | 1177.8 |
| ID * Sex + Age | 704.2 | 1178.2 |
| ID + Sex + Age | 704.5 | 1174.5 |
| ID * Sex | 704.4 | 1174.6 |
| ID + Sex | 704.6 | 1170.9 |
Clearly, adding Age is not warranted here (as is the interaction between ID and Sex)…
As above, we also looked a the simple lmer model:
The compared models are:
| Model | AIC |
|---|---|
| ID * Sex * Age | 1811.2 |
| ID + Sex * Age | 1790.7 |
| ID * Sex + Age | 1786.3 |
| ID + Sex + Age | 1781 |
| ID * Sex | 1777.8 |
| ID + Sex | 1772.5 |
The best model is still the one not including Age:
Linear mixed model fit by REML. t-tests use Satterthwaite's method ['lmerModLmerTest']
Formula: SR ~ 1 + ID + Sex + (1 | Text) + (1 | Speaker) + (1 | Family)
Data: info.rate.data.for.age
REML criterion at convergence: 1758.5
Scaled residuals:
Min 1Q Median 3Q Max
-3.8062 -0.6300 0.0174 0.5882 5.2084
Random effects:
Groups Name Variance Std.Dev.
Speaker (Intercept) 0.35876 0.5990
Text (Intercept) 0.01494 0.1222
Family (Intercept) 0.26668 0.5164
Residual 0.10575 0.3252
Number of obs: 1979, groups: Speaker, 132; Text, 15; Family, 9
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 10.80777 0.77185 27.12524 14.002 6.21e-14 ***
ID -0.70707 0.12477 31.07295 -5.667 3.15e-06 ***
Sex1 -0.14683 0.05268 119.92464 -2.787 0.00618 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) ID
ID -0.971
Sex1 -0.001 0.003Conclusions about Age
Within the limits of our reduced dataset (containing only 132 speakers with Age info), we found the following:
When modelling SR and IR with GAMLSS, while there are hints that Age has, for both, overall:
- a negative main effect (i.e., older speakers of both sexes are slower and transmit less information per unit time across texts and languages), and
- an interaction with Sex (positive for males, i.e., males are less negatively affected by age that females across texts and languages),
it does not seem warranted to include it in these models.
When modelling the relationship between SR and ID, this negative relationship:
- seems strengthened with increasing Age (i.e., older speakers of both sexes shows a stronger negative relationship between SR and ID), and
- there is an interaction with Sex (for females, this effect of Age is stronger than for males),
but, alas, the inclusion of Age is not warranted in the GAMLSS model, nor (really) in the simpler LMER model.
Thus, while Age seems to negatively influence (in a sex-dependent manner) both SR and IR, as well as strengthen the negative relationship between them, its effects are far from clear in the current dataset.
Mixture of Gaussians
In what follows, mixing probabilities are independent from factors such as Sex.
SR
Between 1 and 5 Gaussian distributions:
1 component
Mixing Family: "NO"
Fitting method: EM algorithm
Call: gamlssMX(formula = SR ~ 1, family = NO, K = 1, data = d, plot = FALSE)
Mu Coefficients for model: 1
(Intercept)
6.631
Sigma Coefficients for model: 1
(Intercept)
0.1378
Estimated probabilities: 1
Degrees of Freedom for the fit: 2 Residual Deg. of Freedom 2286
Global Deviance: 7123.61
AIC: 7127.61
SBC: 7139.08 2 components
Mixing Family: c("NO", "NO")
Fitting method: EM algorithm
Call: gamlssMX(formula = SR ~ 1, family = NO, K = 2, data = d, plot = FALSE)
Mu Coefficients for model: 1
(Intercept)
7.149
Sigma Coefficients for model: 1
(Intercept)
-0.1855
Mu Coefficients for model: 2
(Intercept)
5.259
Sigma Coefficients for model: 2
(Intercept)
-0.4788
Estimated probabilities: 0.7260515 0.2739485
Degrees of Freedom for the fit: 5 Residual Deg. of Freedom 2283
Global Deviance: 7001.22
AIC: 7011.22
SBC: 7039.9 3 components
Mixing Family: c("NO", "NO", "NO")
Fitting method: EM algorithm
Call: gamlssMX(formula = SR ~ 1, family = NO, K = 3, data = d, plot = FALSE)
Mu Coefficients for model: 1
(Intercept)
7.106
Sigma Coefficients for model: 1
(Intercept)
-0.2769
Mu Coefficients for model: 2
(Intercept)
7.207
Sigma Coefficients for model: 2
(Intercept)
-0.133
Mu Coefficients for model: 3
(Intercept)
5.27
Sigma Coefficients for model: 3
(Intercept)
-0.4741
Estimated probabilities: 0.3374419 0.3826253 0.2799327
Degrees of Freedom for the fit: 8 Residual Deg. of Freedom 2280
Global Deviance: 7000.8
AIC: 7016.8
SBC: 7062.69 4 components
Mixing Family: c("NO", "NO", "NO", "NO")
Fitting method: EM algorithm
Call: gamlssMX(formula = SR ~ 1, family = NO, K = 4, data = d, plot = FALSE)
Mu Coefficients for model: 1
(Intercept)
7.784
Sigma Coefficients for model: 1
(Intercept)
-0.3468
Mu Coefficients for model: 2
(Intercept)
5.811
Sigma Coefficients for model: 2
(Intercept)
-0.2546
Mu Coefficients for model: 3
(Intercept)
7.124
Sigma Coefficients for model: 3
(Intercept)
-0.7752
Mu Coefficients for model: 4
(Intercept)
5.442
Sigma Coefficients for model: 4
(Intercept)
-0.3356
Estimated probabilities: 0.2584335 0.2204151 0.2985522 0.2225992
Degrees of Freedom for the fit: 11 Residual Deg. of Freedom 2277
Global Deviance: 6990.94
AIC: 7012.94
SBC: 7076.03 5 components
Mixing Family: c("NO", "NO", "NO", "NO", "NO")
Fitting method: EM algorithm
Call: gamlssMX(formula = SR ~ 1, family = NO, K = 5, data = d, plot = FALSE)
Mu Coefficients for model: 1
(Intercept)
6.396
Sigma Coefficients for model: 1
(Intercept)
-0.3948
Mu Coefficients for model: 2
(Intercept)
6.504
Sigma Coefficients for model: 2
(Intercept)
-0.3149
Mu Coefficients for model: 3
(Intercept)
5.132
Sigma Coefficients for model: 3
(Intercept)
-0.5414
Mu Coefficients for model: 4
(Intercept)
7.267
Sigma Coefficients for model: 4
(Intercept)
-1.037
Mu Coefficients for model: 5
(Intercept)
7.959
Sigma Coefficients for model: 5
(Intercept)
-0.4526
Estimated probabilities: 0.2027298 0.180915 0.2175461 0.191181 0.2076281
Degrees of Freedom for the fit: 14 Residual Deg. of Freedom 2274
Global Deviance: 6981.9
AIC: 7009.9
SBC: 7090.2 Comparing AIC
df AIC
mix.SR.NO.5 14 7009.904
mix.SR.NO.2 5 7011.223
mix.SR.NO.4 11 7012.941
mix.SR.NO.3 8 7016.804
mix.SR.NO.1 2 7127.609Showing the distributions
Mixture of Gaussians for SR.
IR
Between 1 and 5 Gaussian distributions:
1 component
Mixing Family: "NO"
Fitting method: EM algorithm
Call: gamlssMX(formula = IR ~ 1, family = NO, K = 1, data = d, plot = FALSE)
Mu Coefficients for model: 1
(Intercept)
39.15
Sigma Coefficients for model: 1
(Intercept)
1.629
Estimated probabilities: 1
Degrees of Freedom for the fit: 2 Residual Deg. of Freedom 2286
Global Deviance: 13945.2
AIC: 13949.2
SBC: 13960.7 2 components
Mixing Family: c("NO", "NO")
Fitting method: EM algorithm
Call: gamlssMX(formula = IR ~ 1, family = NO, K = 2, data = d, plot = FALSE)
Mu Coefficients for model: 1
(Intercept)
41.27
Sigma Coefficients for model: 1
(Intercept)
1.863
Mu Coefficients for model: 2
(Intercept)
38.38
Sigma Coefficients for model: 2
(Intercept)
1.447
Estimated probabilities: 0.266011 0.733989
Degrees of Freedom for the fit: 5 Residual Deg. of Freedom 2283
Global Deviance: 13894.8
AIC: 13904.8
SBC: 13933.5 3 components
Mixing Family: c("NO", "NO", "NO")
Fitting method: EM algorithm
Call: gamlssMX(formula = IR ~ 1, family = NO, K = 3, data = d, plot = FALSE)
Mu Coefficients for model: 1
(Intercept)
40
Sigma Coefficients for model: 1
(Intercept)
0.8341
Mu Coefficients for model: 2
(Intercept)
42.1
Sigma Coefficients for model: 2
(Intercept)
1.725
Mu Coefficients for model: 3
(Intercept)
36.09
Sigma Coefficients for model: 3
(Intercept)
1.395
Estimated probabilities: 0.2444004 0.350947 0.4046526
Degrees of Freedom for the fit: 8 Residual Deg. of Freedom 2280
Global Deviance: 13875.8
AIC: 13891.8
SBC: 13937.6 4 components
Mixing Family: c("NO", "NO", "NO", "NO")
Fitting method: EM algorithm
Call: gamlssMX(formula = IR ~ 1, family = NO, K = 4, data = d, plot = FALSE)
Mu Coefficients for model: 1
(Intercept)
41
Sigma Coefficients for model: 1
(Intercept)
1.127
Mu Coefficients for model: 2
(Intercept)
39.56
Sigma Coefficients for model: 2
(Intercept)
0.08543
Mu Coefficients for model: 3
(Intercept)
36.14
Sigma Coefficients for model: 3
(Intercept)
1.38
Mu Coefficients for model: 4
(Intercept)
42.51
Sigma Coefficients for model: 4
(Intercept)
1.774
Estimated probabilities: 0.222081 0.07674445 0.4389749 0.2621997
Degrees of Freedom for the fit: 11 Residual Deg. of Freedom 2277
Global Deviance: 13868.4
AIC: 13890.4
SBC: 13953.5 5 components
Mixing Family: c("NO", "NO", "NO", "NO", "NO")
Fitting method: EM algorithm
Call: gamlssMX(formula = IR ~ 1, family = NO, K = 5, data = d, plot = FALSE)
Mu Coefficients for model: 1
(Intercept)
40.91
Sigma Coefficients for model: 1
(Intercept)
1.303
Mu Coefficients for model: 2
(Intercept)
34.42
Sigma Coefficients for model: 2
(Intercept)
1.252
Mu Coefficients for model: 3
(Intercept)
39.47
Sigma Coefficients for model: 3
(Intercept)
0.5634
Mu Coefficients for model: 4
(Intercept)
43.68
Sigma Coefficients for model: 4
(Intercept)
1.767
Mu Coefficients for model: 5
(Intercept)
39.64
Sigma Coefficients for model: 5
(Intercept)
1.289
Estimated probabilities: 0.211754 0.2798107 0.1459857 0.1801956 0.182254
Degrees of Freedom for the fit: 14 Residual Deg. of Freedom 2274
Global Deviance: 13868.7
AIC: 13896.7
SBC: 13977 Comparing AIC
df AIC
mix.IR.NO.4 11 13890.37
mix.IR.NO.3 8 13891.76
mix.IR.NO.5 14 13896.71
mix.IR.NO.2 5 13904.80
mix.IR.NO.1 2 13949.21Showing the distributions
Mixture of Gaussians for IR.
Tests of unimodality
We used three ways to estimate how unimodal a distribution is, as they tend to disagree and the problem of unimodality testing is far from settled (see Freeman & Dale, 2013):
- the Silverman test tests the null hypothesis that an underlying density has at most k modes. The null hypothesis is that the underlying density has at most k modes (H0: number of modes <= k). The result is the p-value (bootstrapped) of rejecting a unimodal distribution. It is described in Silverman (1981) and Hall & York (2001); our implementation is based on the code available at https://www.mathematik.uni-marburg.de/~stochastik/R_packages/;
- the dip test which computes Hartigans’ dip statistic D_n_, and its p-value for the test for unimodality, by interpolating tabulated quantiles of sqrt(n) • D_n_. For X_i_ ∼ F, i.i.d., the null hypothesis is that F is a unimodal distribution. The result is the D metric and the p-value (interpolated) of rejecting a unimodal distribution. See Hartigan (1985) and Hartigan & Hartigan (1985); it is implemented in package
diptest; - the bimodality coefficient (BC) which is based on an empirical relationship between bimodality and the third and fourth statistical moments of a distribution (skewness and kurtosis). It is proportional to the division of squared skewness with uncorrected kurtosis, BC ≈ (s2 + 1)/k, with the underlying logic that a bimodal distribution will have very low kurtosis, an asymmetric character, or both; all of these conditions increase BC. The values range from 0 and 1, with those exceeding .555 (the value representing a uniform distribution) suggesting bi-modality. The result is the DC estimate (which must exceed 0.555 to reject a unimodal distribution). We implemented it following Freeman & Dale (2013) as BC = (s2 + 1)/(k + 3 • ((n-1)2 / ((n-2) • (n-3)))).
For each such test, we performed four randomisation procedures to obtain an estimate of the “specialness” of the observed unimodality estimate; for each new permuted dataset, we recompute everything before estimating the unimodlaity of the permuted distribution:
- Permutation model 1 (PM1): randomly permute the SR values freely among speakers, texts and languages;
- Permutation model 2 (PM2): randomly permute the ID values among languages;
- Permutation model 3 (PM3): randomly permute the Speaker average SR values among speakers (irrespective of languages);
- Permutation model 4 (PM4): randomly permute the Language average SR values among languages.
Visual comparison
The observed estimate (vertical blue solid line), the permuted distribution (gray histogram), and the “unimodality region” (shaded green rectangle) are shown below (for PM3, we also show the original estimate using the Speaker average SR as a vertical solid red line).
PM1
Permutation of the texts’ SRs (PM1).
PM2
Permutation of the languages’ ID (PM2).
PM3
Permutation of the speakers’ average SRs (PM3).
PM4
Permutation of the languages’ average SRs with speaker adjustement (PM4).
Summary of unimodality tests
| Scenario | Measure | Test | Observed estimate (p-value) | % more unimodal permutations |
|---|---|---|---|---|
| PM1 | SR | Silverman | - (0.017) | 98% |
| PM1 | SR | Dip | 0.005 (0.984) * | 100% |
| PM1 | SR | BC | 0.19 () * | 100% |
| PM1 | IR | Silverman | - (0.85) * | 13.1% |
| PM1 | IR | Dip | 0.005 (0.992) * | 73.3% |
| PM1 | IR | BC | 0.167 () * | 100% |
| PM2 | SR | Silverman | - (0.017) | 98% |
| PM2 | SR | Dip | 0.005 (0.984) * | 100% |
| PM2 | SR | BC | 0.19 () * | 100% |
| PM2 | IR | Silverman | - (0.85) * | 1.9% |
| PM2 | IR | Dip | 0.005 (0.992) * | 26.5% |
| PM2 | IR | BC | 0.167 () * | 96.7% |
| PM3 | SR | Silverman | - (0.017) | 100% |
| PM3 | SR | Dip | 0.005 (0.984) * | 0% |
| PM3 | SR | BC | 0.19 () * | 100% |
| PM3 | IR | Silverman | - (0.85) * | 14% |
| PM3 | IR | Dip | 0.005 (0.992) * | 18.4% |
| PM3 | IR | BC | 0.167 () * | 99.9% |
| PM4 | SR | Silverman | - (0.017) | 6.6% |
| PM4 | SR | Dip | 0.005 (0.984) * | 1.8% |
| PM4 | SR | BC | 0.19 () * | 82.8% |
| PM4 | IR | Silverman | - (0.85) * | 0.9% |
| PM4 | IR | Dip | 0.005 (0.992) * | 14.6% |
| PM4 | IR | BC | 0.167 () * | 96.3% |
Pair-wise distances between languages
We compute various distances between languages (as implemented by function distance() in package philentropy) in what concerns the distribution of NS, SR and ID.
Comparing the distribution of pairwise distances between languages.
| m1 | m2 | d | mean1 | median1 | sd1 | mean2 | median2 | sd2 | p |
|---|---|---|---|---|---|---|---|---|---|
| IR | NS | Hellinger | 0.88 | 0.83 | 0.32 | 1.20 | 1.19 | 0.39 | 0.00 |
| IR | NS | Jensen-Shannon | 0.17 | 0.14 | 0.12 | 0.29 | 0.26 | 0.16 | 0.00 |
| IR | NS | Kolmogorov–Smirnov | 0.42 | 0.37 | 0.20 | 0.57 | 0.57 | 0.23 | 0.00 |
| IR | NS | Kullback-Leibler | 7.13 | 4.22 | 7.81 | 15.42 | 13.12 | 13.31 | 0.00 |
| IR | NS | Squared-Chi | 0.56 | 0.45 | 0.36 | 0.88 | 0.79 | 0.47 | 0.00 |
| IR | SR | Hellinger | 0.88 | 0.83 | 0.32 | 1.10 | 1.06 | 0.48 | 0.00 |
| IR | SR | Jensen-Shannon | 0.17 | 0.14 | 0.12 | 0.27 | 0.23 | 0.20 | 0.00 |
| IR | SR | Kolmogorov–Smirnov | 0.42 | 0.37 | 0.20 | 0.56 | 0.55 | 0.27 | 0.00 |
| IR | SR | Kullback-Leibler | 7.13 | 4.22 | 7.81 | 12.80 | 6.69 | 14.88 | 0.00 |
| IR | SR | Squared-Chi | 0.56 | 0.45 | 0.36 | 0.86 | 0.77 | 0.57 | 0.00 |
| NS | SR | Hellinger | 1.20 | 1.19 | 0.39 | 1.10 | 1.06 | 0.48 | 0.00 |
| NS | SR | Jensen-Shannon | 0.29 | 0.26 | 0.16 | 0.27 | 0.23 | 0.20 | 0.32 |
| NS | SR | Kolmogorov–Smirnov | 0.57 | 0.57 | 0.23 | 0.56 | 0.55 | 0.27 | 0.79 |
| NS | SR | Kullback-Leibler | 15.42 | 13.12 | 13.31 | 12.80 | 6.69 | 14.88 | 0.04 |
| NS | SR | Squared-Chi | 0.88 | 0.79 | 0.47 | 0.86 | 0.77 | 0.57 | 0.60 |
References
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Freeman, J. B., & Dale, R. (2013). Assessing bimodality to detect the presence of a dual cognitive process. Behavior research methods, 45(1), 83-97.
Hall, P., & York, M. (2001). On the calibration of Silverman’s test for multimodality. Statistica Sinica, 11, 515-536.
Hartigan, J. A., & Hartigan, P. M. (1985) The Dip Test of Unimodality. Annals of Statistics 13, 70–84.
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Appendix I: R session info
This document was compiled on:
R version 3.4.4 (2018-03-15)
**Platform:** x86_64-pc-linux-gnu (64-bit)
locale: LC_CTYPE=en_US.UTF-8, LC_NUMERIC=C, LC_TIME=en_US.UTF-8, LC_COLLATE=en_US.UTF-8, LC_MONETARY=en_US.UTF-8, LC_MESSAGES=en_US.UTF-8, LC_PAPER=en_US.UTF-8, LC_NAME=C, LC_ADDRESS=C, LC_TELEPHONE=C, LC_MEASUREMENT=en_US.UTF-8 and LC_IDENTIFICATION=C
attached base packages: grid, parallel, splines, stats, graphics, grDevices, datasets, utils, methods and base
other attached packages: ggpubr(v.0.2), magrittr(v.1.5), ggrepel(v.0.8.0), RColorBrewer(v.1.1-2), dplyr(v.0.8.0.1), ggridges(v.0.5.1), broman(v.0.68-2), philentropy(v.0.3.0), pander(v.0.6.3), moments(v.0.14), sjPlot(v.2.6.2), sjstats(v.0.17.3), gamlss.mx(v.4.3-5), nnet(v.7.3-12), gamlss(v.5.1-2), nlme(v.3.1-137), gamlss.dist(v.5.1-1), MASS(v.7.3-51.1), gamlss.data(v.5.1-0), lmerTest(v.3.1-0), lme4(v.1.1-20), Matrix(v.1.2-15), plyr(v.1.8.4), reshape2(v.1.4.3), ggplot2(v.3.1.0) and RhpcBLASctl(v.0.18-205)
loaded via a namespace (and not attached): numDeriv(v.2016.8-1), tools(v.3.4.4), TMB(v.1.7.15), backports(v.1.1.3), R6(v.2.4.0), sjlabelled(v.1.0.16), lazyeval(v.0.2.1), colorspace(v.1.4-0), withr(v.2.1.2), tidyselect(v.0.2.5), mnormt(v.1.5-5), emmeans(v.1.3.2), compiler(v.3.4.4), sandwich(v.2.5-0), labeling(v.0.3), scales(v.1.0.0), mvtnorm(v.1.0-8), psych(v.1.8.12), stringr(v.1.4.0), digest(v.0.6.18), foreign(v.0.8-71), minqa(v.1.2.4), rmarkdown(v.1.11), stringdist(v.0.9.5.1), pkgconfig(v.2.0.2), htmltools(v.0.3.6), highr(v.0.7), pwr(v.1.2-2), rlang(v.0.3.1), generics(v.0.0.2), zoo(v.1.8-4), modeltools(v.0.2-22), bayesplot(v.1.6.0), Rcpp(v.1.0.0), munsell(v.0.5.0), prediction(v.0.3.6.2), stringi(v.1.3.1), multcomp(v.1.4-8), yaml(v.2.2.0), snakecase(v.0.9.2), sjmisc(v.2.7.7), forcats(v.0.4.0), crayon(v.1.3.4), lattice(v.0.20-38), ggeffects(v.0.8.0), haven(v.2.1.0), hms(v.0.4.2), knitr(v.1.21), pillar(v.1.3.1), estimability(v.1.3), codetools(v.0.2-16), stats4(v.3.4.4), glue(v.1.3.0), evaluate(v.0.13), data.table(v.1.12.0), modelr(v.0.1.4), nloptr(v.1.2.1), gtable(v.0.2.0), purrr(v.0.3.0), tidyr(v.0.8.2), assertthat(v.0.2.0), xfun(v.0.5), coin(v.1.2-2), xtable(v.1.8-3), broom(v.0.5.1), coda(v.0.19-2), survival(v.2.43-3), tibble(v.2.0.1), glmmTMB(v.0.2.3) and TH.data(v.1.0-10)
Appendix II: Figures for the main paper
Here we generate the figures used in the main paper (saved to the ./figures folder as 600 DPI TIFF files Figure-*.tiff).